Boundary complexification: 3D, 4D, and more

Reimagining Guernica to Engage the Antitheses of a Cancel Culture (Part #4)

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Fields and plans: It is too readily assumed that boundaries are simply delineated -- defined by a simple line. This simplification extends to the "fields" which are thereby defined. The pattern is evident in the image of standard tennis court (below left). Countries are defined by an elaboration of that principle. The pattern is extended by implication to fields of knowledge, as cultivated by distinct disciplines -- anxious to protect their distinctive identity. The implied two dimensionality is also evident in planning and the distinctive organization of institutional responsibilities.

Boundary relationships
2D: Tennis court Conventional ball-game boundaries	3D: Contiguity mapping with Szilassi polyhedron Schematic of Commonwealth of Australia	Changing boundary rules between states of Commonwealth of Australia
NielsF, CC BY-SA 3.0, via Wikimedia Commons

Goals and holes: Curiously there is an implicit three dimensional challenge to such planar understanding  notably highlighted in many ball games in which the objective is to place the ball in a goal. This is typically a net which is outside the plane of fields. Any kind of goal could be considered in this light.

More intriguing is the sense in which the goal can be understood as taking the form of a hole -- with a field itself potentially to be seen in this way, especially if it is a target vulnerable to invasion by an "outsider".

Configurations of fields, holes and goals: Whereas a tennis court is an example of a planar configuration of fields, geometry indicates far more complex possibilities, as extensively discussed separately (Engaging with Globality -- through cognitive lines, circlets, crowns or holes, 2009).

In such a context, it is appropriate to note the relation between field and court with the latter featuring as a tennis court, in a court of law, and the court of public opinion -- and especially given their interplay in the Djokovic saga, then readily understood as a "three ring circus" (Emma John, The Djokovic circus allows us to see all our Covid prejudices being played out, The Guardian, 16 January 2022). Also intriguing is recognition of the process of "shifting the goal posts" in any planning process.

Szilassi polyhedron: Of some relevance to this argument in relation to the Commonwealth of Australia is its configuration as a federation of seven states. These can of course be presented as a conventional two-dimensional map. This obscures any sense of their relationship to each other in the federation through which the Commonwealth is understood as a whole. It is therefore of potential interest to note the unique property of a Szilassi polyhedron in 3D as shown above. It is constructed from seven hexagonal faces with the unique topological quality that each face within the polyhedron directly touches every other face within the 3D configuration -- as do the the states of the federation in principle.

The configuration is highly unusual (as depicted above centre) -- taking a toroidal form -- with the central hole indicative of the Australian Capital Territory (Canberra). Although seemingly obscure, the Szilassi polyhedron figures as one of the works of monumental art -- with the Mobius Strip -- in Reconciliation Place (Canberra). This is located between the National Library and High Court of Australia, as a tribute to the Indigenous people of Australia.

The animation (above right) offers an indication of the border changes between states in response to the pandemic, of which 61 were noted in the case of Queensland.

Trajectories and knots: Any possible complexification can be explored otherwise through the movement of a ball in relation to such a configuration. Given the insights of the Uncertainty Principle, there is an argument for interrelating the elements of the geometry: does the ball trace out boundary lines and/or frame invasion of a field? When is a field a goal (to be invaded) or when is a goal to be understood as a field (to be explored)? Such questions are indicative of a sense in which a ball-game can be understood as a knot, as explored by topology.

The relevance to governance of the knot meme can be variously discussed. For example, it is used by Michael Lesher in introducing discussion of the current problematic prevalence of lies:

Remember the good old days at the beginning of the COVID coup â-- before the new dogmas started to tangle themselves into self-contradictory knots? (Truth or Covid? (or, â-"e;why we know everything they're telling us is a lieâ-") OffGuardian, 28 January 2022)

The images below offer contrasting examples of how fields in a configuration may be entangled when the ball has a trajectory been one court and another.

Commenting on a session of the World Economic Forum, John Jullens argues that: It's as if the global economy is being strangled by a gigantic Gordian knot from which it cannot untangle itself (The Gordian Knot of Global Economic Growth, Strategy-Business, 15 October 2013). In the absence of depictions of such a knot in cognitive terms, the implication that the dilemmas of global management might be explored topologically as a knot merits consideration in the light of the mathematical interest in the endless knot, the trefoil knot, the cinquefoil knot, and the septfoil knot (as shown below).

Challenges to imagining toroidal life? (Reproduced from Imagining Toroidal Life as a Sustainable Alternative From Globalization to Toroidization or back to Flatland?2019)
Trefoil knot	Cinquefoil knot	Septfoil knot	Seifert surface bounded by a set of Borromean rings
By Jim.belk Animation: MichaelFrey (talk) - Own work, Public Domain, Link	By Jim.belk - Own work, Public Domain, Link	By Jim.belk - Own work, Public Domain, Link	Reproduced from Wikipedia [more images]

This would be consistent with the psychological significance associated with knot topology by Jacques Lacan and R. D, Laing -- in respect of individual "self-governance". (R. D. Laing, Knots, 1972; Jean Michel Vappereau, Knot: the theory of the knot outlined by Jacques Lacan, Lacanian Works, July 1996). Especially intriguing with respect to the higher dimensionality of string theory, is the choice of "string" as a metaphor, given the contrasting importance attached to "knot" as a metaphor by psychoanalysis.

Borromean interlocking of fields: The Borromean ring configuration is of great significance in logic and mathematics. It has been chosen as the logo of the International Mathematical Union, as in the design by John M. Sullivan (New IMU Logo based on the tight Borromean rings, 2006; Charles Gunn and John M. Sullivan, The Borromean Rings: a video about the new IMU Logo, 2008). Their relevance to the coordination of socio-political systems is typically ignored, whatever role they may have in dialogue informed by symbolic dimensions.

Examples of 3-fold articulations of Borromean rings of relevance to coordination of political systems? (Reproduced from Engaging with Elusive Connectivity and Coherence Global comprehension as a mistaken quest for closure, 2018)
Representation of rings interlocking according to the Borromean condition
Early depiction of Christian Trinity	Common representation in 2D	Distorted representation as ellipses	3D representation	International Mathematical Union Logo
Reproduced from Wikipedia

Knot tables and ball-related trajectories: As noted above, the relation between knots and cognitive processes has been a concern to psychiatrists. Laing expanded the view of the â-"e;double bindâ-" hypothesis put forth by Gregory Bateson and his colleagues, formulating a new concept to describe the highly complex situation that unfolds in the process of "going mad", namely an "incompatible knot". Arguably global society is currently in a condition which merits recognition as a form of knot inhibiting meaningful dialogue. Knot theory and knot tables (as shown below) are potentially relevant to the possibility (Cognitive embodiment of knots: knotting and knitting processes, 2021).

Of particular interest is the "knot" within which a tennis ball or baseball can be understood to be wrapped. As depicted below, it is potentially indicative of a form of self-reflexivity and self-penetration -- of the kind on which psychiatrists have focused (Globalization: playing ball, self-reflexivity and self-penetration? 2021). Speculation can be taken further with reference to widespread use of the baseball cap (Correspondence between the baseball curve and the baseball cap? 2020; Baseball Cap Implications in the Quest for Global Hegemony: comprehension of elusive order through the dynamics of angels and demons, 2020).

Knots and ball seam curves
(Reproduced from Dynamics of N-fold Integration of Disparate Cognitive Modalities , 2021)
Table of all prime knots with up to seven crossings represented as knot diagrams with their medial graph.	Baseball / Tennis ball seam curve	Baseball cap in relation to seam curve
User:Jkasd and User:Christian.Mercat, Public domain, via Wikimedia Commons	Interactive 3D variant

Minimal knot of requisite complexity? If a knot or juggling pattern could be understood as "holding" the requisite variety of cognitive operations for a viable system,  there is a case for exploring a recently discovered knot which has been the focus of extensive attention. This is most succinctly presented by Louis Kauffman (Pattern, Sign and Space: Mereon Thoughts. University of Illinois at Chicago, 2003). Otherwise known and visualized as the Mereon Matrix and the Mereon Trefoil, its potential significance is elaborated in a far more extensive work (Louis H Kauffman, et al, The Mereon Matrix: everything connected through (k)nothing, 2018; frontmatter) to which detailed reference is made in the conclusion of a related exploration (Identifying Polyhedra Enabling Memorable Strategic Mapping, 2020).

Animations offering contrasting perspectives on the Mereon Trefoil pattern (Adapted from variants in Dynamics of N-fold Integration of Disparate Cognitive Modalities , 2021)
Interactive version in X3DOM Animations adapted from X3D models kindly produced by Sergey Bederov of Cortona3D.

Of particular interest is use of the knot as a container for the movement of a ball along its length as shown below. This helps to indicate the ambiguity by which movement of the ball both defines the knot and "invades" the fields "delineated" by the interlocking loops of the knot. The significance of ball and field is then entangled in  a manner indicative of that between field and goal. This offers ways of reflecting on the challenges of inter-disciplinary initiatives in which the disciplinary fields are entangled and subject to mutual "invasion" -- as with their inter-national and inter-faith analogues.

Screenshots of wireframe renderings of animations immediately above (with inclusion of moving ball)
Interactive version in X3DOM (click there for wireframe rendering)

The sense of invasion as self-penetration is discussed separately (Globalization: playing ball, self-reflexivity and self-penetration?2021). There is the intriguing possibility that each loop is a partial approximation to an incomplete torus -- then to be understood as framing a disk in process of being twisted. The surface then bears comparison with a Seifert surface (as depicted above). The trefoil might then be understood as a static condition of a set of several spinning disks, each of the kind described as an Euler's disk, but with a different axis of spin,

Quantum paradigm shift: The conventional focus on the static nature of fields is currently challenged by the cognitive implications of the ongoing shift to a quantum perspective. Related insights have been extensively developed and clarified with respect to international relations by Alexander Wendt (Quantum Mind and Social Science: unifying physical and social ontology, 2015; The mind-body problem and social science: motivating a quantum social theory, Journal for the Theory of Social Behaviour, 48, 2018, 2). Given the implications for governance, of further relevance is Wendt's provocative argument that individuals merit recognition as "walking wave functions", as discussed separately (On being "walking wave functions" in terms of quantum consciousness? 2017).

With respect to conventional boundaries, Wendt notes that constructs like nation states (as with "USA", "Russia", or "Australia") are effectively legal fictions -- especially from any hypothetical extraterrestrial perspective. The argument can be readily extended to the fields configured by science and religion and the nature of their "existence". Citing Wendt, David Orrell asks when will economists let go of physics envy to embrace the quantum revolution? (A Softer Economics, Aeon, 1 February 2022). His critique offers valuable insight into the problematic responses of both physicists and social scientists to the unconventional insights offered from a quantum perspective.

Animations of the movement of a ball within the knot (such as those above) are potentially helpful to reflection on the paradoxical quantum implications of superposition and entanglement -- especially if the number of circulating balls is increased and they pass through each other.

Laws of Form: In the midst of the Djokovic saga the prime minister of Australia was constrained to make the seemingly fundamental assertion that Rules are Rules (Andrew Reid, 'Rules are rules': Scott Morrison hits out after Djokovic bombshell, Yahoo Sports, 6 January 2022; Jonathan Talbot, 'No one is above the rules': Scott Morrison responds to Novak Djokovic visa rejection at Melbourne Airport, Sky News, 6 January 2022). The assertion frames the question as to whether it constitutes a rule in its own right. The irony has been variously noted, as by Callum Foote (Yes, â-"e;rules are rulesâ-" Scott, you invent them, break them, subvert them and ignore them 14 January 2022):

PM Scott Morrison came out strongly against Novak Djokovic again this week, claiming "rules are rules â-... no one is above these rules". Yet the list of this government's broken rules is prolific.Â

Part of the problem seems to lie in a missing link in the relation of mathematics to logic which has been provided, with the encouragement of Bertrand Russell, by George Spencer Brown (Laws of Form, 1969). As discussed separately, he argues that: nobody hitherto appears to have made any sustained attempt to elucidate and to study the primary, non-numerical arithmetic of the algebra in everyday use which now bears Boole's name. And again:

That mathematics, in common with other art forms, can lead us beyond ordinary existence, and can show us something of the structure in which all creation hangs together, is no new idea. But mathematical texts generally begin the story somewhere in the middle, leaving the reader to pick up the thread as best he can. Here the story is traced from the beginning.

The result of Spencer Brown's formal exercise to separate what are known as algebras of logic from the subject of logic, and to re-align them with mathematics is the explicit, and extremely elegant logical re-integration of the observer. His final chapter, entitled "reentry into the form" commences with:

The conception of the form lies in the desire to distinguish. Granted this desire, we cannot escape the form, although we can see it any way we please (p. 69).

It ends with:

An observer, since he distinguishes the space he occupies, is also a mark... In this conception a distinction drawn in any space is a mark distinguishing the space. Equally and conversely, any mark in a space draws a distinction. We see now that the first distinction, the mark, and the observer are not only interchangeable, but, in the form, identical. (p. 76)

(Daniel G. Schwartz, Isomorphisms of G. Spencer-Brown's Laws of Form and F. Varela's Calculus for Self-Reference . International Journal of General Systems.6, 1981, 4)

Unsayable and unsaid: The challenge of thriving in this cognitive environment is then less a question of locating relevant literature, learning the knowledge it contains, or citing it to justify positions to others. Nor is it a question of who has been there before, or any criticism from some other perspective of "rediscovering the wheel". The question might even be the validity of the external frame from which that question could be asked. By whom is one to be persuaded, about what and why -- and why should one seek to persuade? To what extent is any essential incommunicability a matter of Ludwig Wittgenstein's concluding phrase: Whereof one cannot speak, thereof one must be silent (Tractatus Logico-Philosophicus, 1921).

As discussed previously (Evolutionary influence of the absent, 2011), with respect to the argument of Terrence W. Deacon (Incomplete Nature: how mind emerged from matter, 2011), a key factor with respect to the emergence of knowledge may be intimately associated with what is missing -- a point succinctly made in the contrast between the print and online summaries of his argument (The importance of what is missing, New Scientist, 26 November 2011; Consciousness is a matter of constraint, New Scientist, 30 November 2011). For Deacon:

... have we been looking in the wrong places for clues? ... brain researchers and philosophers of mind have focused on brain processes, neural computations and their correspondences with the material world. But what if we should be focusing on what is not there instead? ... I believe that in order to overcome this stalemate we need to pay more attention to what is intrinsically not present in everything -- from life's functions and meanings to mind's experiences and values. [emphasis added]

Openness and closure: There is an interesting sense in which geometric metaphors are about enclosure of "space" -- as with "field"-- are themselves significant according to the epistemological arguments of Hilary Lawson (Closure: A Short History of Everything, 2002). The aspiration of physicists could be seen as a device to enclose space and time such as to preclude any new thought on the matter -- reminiscent of forms of "lock-in" and escalation of commitment (familiar in decision-making processes), technological lock-in, or customer lock-in (familiar as a marketing strategy).

The complementary roles of closing and opening have been explored by Orrin E. Klapp (Opening and Closing; strategies and information adaptation in society, 1978), as discussed separately (Openness and Closure in Pattern Language, 2012) . Much has been made in recent years of an "open society" and of an "open source" approach to knowledge management -- in contrast with previous enthusiasm for closure, and continuing advocacy thereof (as epitomized by ACTA).

With respect to one major strategic response to the pandemic, there is a certain irony to the focus on limiting "close contact" through social distancing as a restrictive response to the appeal of openness. ***

Possession: It is curious to note the extent to which ball games -- and their analogues with respect to any bounded field or domain -- typically involve a highly developed sense of exclusive possession. A field of preoccupation of any kind readily evokes an overriding sense of property -- as with control of any ball (or its analogue) within it. It invites comparison with the defence of a goal. In the case of the pandemic, this has been especially problematic in relation to vaccine patents -- held restrictively irrespective of any fatal implications (and the hole engendered in national budgets).

The issue is especially evident with respect to the cultural property that is the preoccupation of academic disciplines, religions and political ideologies. It can be understood as an unspoken characteristic of the innovation promoted through the Triple helix model of innovation and the focus on copyright (Future Coping Strategies: beyond the constraints of proprietary metaphors, 1992). There is considerable irony to the possibility that the relationship between fields as the frames of on which Einstein focused avoids a more fundamental issue of how they are "possessed" (Einstein's Implicit Theory of Relativity - of Cognitive Property? Unexamined influence of patenting procedures, 2007).

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